Auto-Validating Numerical Methods

The main goal of this text is to introduce the reader to the field of auto-validating algorithms by providing a theoretical foundation supplemented with illuminating examples. The target audience is undergraduate or graduate students new to the field. Some knowledge of programming is useful, but not strictly necessary. The restriction to the one-dimensional setting is a conscious one. It allows us to focus exclusively on simple, but yet interesting problems, without too much mathematical framework. The computer exercises are intended to "force" the reader into actually discovering how simple it is to implement the methods, and to solve numerical problems with rigour. TOC:Introduction.- 1 Computer arithmetic: 1.1 Positional systems.- 1.2 Floating point numbers .- 1.3 Rounding.- 1.4 Floating point arithmetic.- 1.5 The IEEE standards.- 1.6 Examples of floating point computations.- 1.7 Computer Lab II.- 2 Interval arithmetic: 2.1 Real intervals.- 2.2 Real interval arithmetic.- 2.3 Extended interval arithmetic.- 2.4 Containment sets.- 3 Interval analysis: 3.1 Interval functions.- 3.2 Centered forms.- 3.3 Monotonicity.- 3.4 Computer Lab II.- 4 Automatic differentiation: 4.1 First-order derivatives.- 4.2 Higher-order derivatives.- 4.3 Higher-order enclosures.- Computer Lab III.- 5 Interval analysis in action: 5.1 Root finding.- 5.2 Optimization.- 5.3 Quadrature.- 5.4 Computer Lab IV.- 6 Ordinary differential equations: 6.1 A gentle mathematical introduction.- 6.2 Simple enclosure methods.- 6.3 Higher-order methods.- 6.4 Rigorous higher-order examples.- Appendices: A Mathematical foundations: A.1 The rational numbers.- A.2 What is a real number?- A.3 Completeness.- A4 Fixed point theorems.- B Program codes: B.1 IEEE constants.- B.2 Changing rounding modes.- B.3 IA sample code in C.- B.4 IA sample code in C++.- Bibliography.